#202857

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,10,14)(7,11,16,20)(8,13,9,18)(12,17,19,15)(22,26)(23,30)(24,27)(28,35)(29,33)(31,34);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20)(21,22,27,33,25,29,24,26)(23,28,34,36,31,35,30,32);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19)(21,23,24,30,25,31,27,34)(22,28,26,32,29,35,33,36);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,12,20,19)(22,29)(26,33)(28,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110053);